Lower Semicontinuity of Quasi-convex Bulk Energies in SBV and Integral Representation in Dimension Reduction
Jean‐François Babadjian · SIAM Journal on Mathematical Analysis · 2008
A result of Larsen concerning the structure of the approximate gradient of certain sequences of functions with bounded variation is used to present a short proof of Ambrosio's lower semicontinuity theorem for quasi-convex bulk energies in $SBV$. It enables us to generalize to the $SBV$ setting the decomposition lemma for scaled gradients in dimension reduction and also to show that, from the point of view of bulk energies, $SBV$ dimensional reduction problems can be reduced to analog ones in the Sobolev spaces framework.